2 Equations of General Relativistic Numerical Hydrodynamics in General RelativityNumerical Hydrodynamics in General Relativity

1 Introduction 

The description of important areas of modern astronomy, such as high-energy astrophysics or gravitational wave astronomy, requires general relativity. High-energy radiation is often emitted by highly relativistic events in regions of strong gravitational fields near compact objects such as neutron stars or black holes. The production of relativistic radio jets in active galactic nuclei, explained by pure hydrodynamical effects as in the twin-exhaust model [35], by hydromagnetic centrifugal acceleration as in the Blandford-Payne mechanism [34Jump To The Next Citation Point In The Article], or by electromagnetic extraction of energy as in the Blandford-Znajek mechanism [36Jump To The Next Citation Point In The Article], involves an accretion disk around a rotating supermassive black hole. The discovery of kHz quasi-periodic oscillations in low-mass X-ray binaries extended the frequency range over which these oscillations occur into timescales associated with the relativistic, innermost regions of accretion disks (see, e.g., [288Jump To The Next Citation Point In The Article]). A relativistic description is also necessary in scenarios involving explosive collapse of very massive stars (tex2html_wrap_inline4668) to a black hole (in the so-called collapsar and hypernova models), or during the last phases of the coalescence of neutron star binaries. These catastrophic events are believed to exist at the central engine of highly energetic tex2html_wrap_inline4670 -ray bursts (GRBs) [215, 201, 307Jump To The Next Citation Point In The Article, 216]. In addition, non-spherical gravitational collapse leading to black hole formation or to a supernova explosion, and neutron star binary coalescence are among the most promising sources of detectable gravitational radiation. Such astrophysical scenarios constitute one of the main targets for the new generation of ground-based laser interferometers, just starting their gravitational wave search (LIGO, VIRGO, GEO600, TAMA) [286, 206Jump To The Next Citation Point In The Article].

A powerful way to improve our understanding of the above scenarios is through accurate, large scale, three-dimensional numerical simulations. Nowadays, computational general relativistic astrophysics is an increasingly important field of research. In addition to the large amount of observational data by high-energy X- and tex2html_wrap_inline4670 -ray satellites such as Chandra, XMM-Newton, or INTEGRAL, and the new generation of gravitational wave detectors, the rapid increase in computing power through parallel supercomputers and the associated advance in software technologies is making possible large scale numerical simulations in the framework of general relativity. However, the computational astrophysicist and the numerical relativist face a daunting task. In the most general case, the equations governing the dynamics of relativistic astrophysical systems are an intricate, coupled system of time-dependent partial differential equations, comprising the (general) relativistic (magneto-)hydrodynamic (MHD) equations and the Einstein gravitational field equations. In many cases, the number of equations must be augmented to account for non-adiabatic processes, e.g., radiative transfer or sophisticated microphysics (realistic equations of state for nuclear matter, nuclear physics, magnetic fields, and so on).

Nevertheless, in some astrophysical situations of interest, e.g., accretion of matter onto compact objects or oscillations of relativistic stars, the ``test fluid'' approximation is enough to get an accurate description of the underlying dynamics. In this approximation the fluid self-gravity is neglected in comparison to the background gravitational field. This is best exemplified in accretion problems where the mass of the accreting fluid is usually much smaller than the mass of the compact object. Additionally, a description employing ideal hydrodynamics (i.e., with the stress-energy tensor being that of a perfect fluid), is also a fairly standard choice in numerical astrophysics.

The main purpose of this review is to summarize the existing efforts to solve numerically the equations of (ideal) general relativistic hydrodynamics. To this aim, the most important numerical schemes will be presented first in some detail. Prominence will be given to the so-called Godunov-type schemes written in conservative form. Since [163Jump To The Next Citation Point In The Article], it has been demonstrated gradually [93Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article, 244Jump To The Next Citation Point In The Article, 83, 21Jump To The Next Citation Point In The Article, 297Jump To The Next Citation Point In The Article, 229Jump To The Next Citation Point In The Article] that conservative methods exploiting the hyperbolic character of the relativistic hydrodynamic equations are optimally suited for accurate numerical integrations, even well inside the ultrarelativistic regime. The explicit knowledge of the characteristic speeds (eigenvalues) of the equations, together with the corresponding eigenvectors, provides the mathematical (and physical) framework for such integrations, by means of either exact or approximate Riemann solvers.

The article includes, furthermore, a comprehensive description of ``relevant'' numerical applications in relativistic astrophysics, including gravitational collapse, accretion onto compact objects, and hydrodynamical evolution of neutron stars. Numerical simulations of strong-field scenarios employing Newtonian gravity and hydrodynamics, as well as possible post-Newtonian extensions, have received considerable attention in the literature and will not be covered in this review, which focuses on relativistic simulations. Nevertheless, we must emphasize that most of what is known about hydrodynamics near compact objects, in particular in black hole astrophysics, has been accurately described using Newtonian models. Probably the best known example is the use of a pseudo-Newtonian potential for non-rotating black holes that mimics the existence of an event horizon at the Schwarzschild gravitational radius [217Jump To The Next Citation Point In The Article]. This has allowed accurate interpretations of observational phenomena.

The organization of this article is as follows: Section  2 presents the equations of general relativistic hydrodynamics, summarizing the most relevant theoretical formulations that, to some extent, have helped to drive the development of numerical algorithms for their solution. Section  3 is mainly devoted to describing numerical schemes specifically designed to solve nonlinear hyperbolic systems of conservation laws. Hence, particular emphasis will be paid on conservative high-resolution shock-capturing (HRSC) upwind methods based on linearized Riemann solvers. Alternative schemes such as smoothed particle hydrodynamics (SPH), (pseudo-)spectral methods, and others will be briefly discussed as well. Section  4 summarizes a comprehensive sample of hydrodynamical simulations in strong-field general relativistic astrophysics. Finally, in Section  5 we provide additional technical information needed to build up upwind HRSC schemes for the general relativistic hydrodynamics equations. Geometrized units (G = c =1) are used throughout the paper except where explicitly indicated, as well as the metric conventions of [186]. Greek (Latin) indices run from 0 to 3 (1 to 3).

2 Equations of General Relativistic Numerical Hydrodynamics in General RelativityNumerical Hydrodynamics in General Relativity

image Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
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